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Theorem nnmord 7866
Description: Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnmord ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))

Proof of Theorem nnmord
StepHypRef Expression
1 nnmordi 7865 . . . . . 6 (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
21ex 397 . . . . 5 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
32com23 86 . . . 4 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴𝐵 → (∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵))))
43impd 396 . . 3 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
543adant1 1124 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
6 ne0i 4069 . . . . . . . 8 ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐶 ·𝑜 𝐵) ≠ ∅)
7 nnm0r 7844 . . . . . . . . . 10 (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅)
8 oveq1 6800 . . . . . . . . . . 11 (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
98eqeq1d 2773 . . . . . . . . . 10 (𝐶 = ∅ → ((𝐶 ·𝑜 𝐵) = ∅ ↔ (∅ ·𝑜 𝐵) = ∅))
107, 9syl5ibrcom 237 . . . . . . . . 9 (𝐵 ∈ ω → (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = ∅))
1110necon3d 2964 . . . . . . . 8 (𝐵 ∈ ω → ((𝐶 ·𝑜 𝐵) ≠ ∅ → 𝐶 ≠ ∅))
126, 11syl5 34 . . . . . . 7 (𝐵 ∈ ω → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
1312adantr 466 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐶 ≠ ∅))
14 nnord 7220 . . . . . . . 8 (𝐶 ∈ ω → Ord 𝐶)
15 ord0eln0 5922 . . . . . . . 8 (Ord 𝐶 → (∅ ∈ 𝐶𝐶 ≠ ∅))
1614, 15syl 17 . . . . . . 7 (𝐶 ∈ ω → (∅ ∈ 𝐶𝐶 ≠ ∅))
1716adantl 467 . . . . . 6 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶𝐶 ≠ ∅))
1813, 17sylibrd 249 . . . . 5 ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
19183adant1 1124 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → ∅ ∈ 𝐶))
20 oveq2 6801 . . . . . . . . . 10 (𝐴 = 𝐵 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))
2120a1i 11 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 → (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵)))
22 nnmordi 7865 . . . . . . . . . 10 (((𝐴 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵𝐴 → (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
23223adantl2 1172 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐵𝐴 → (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)))
2421, 23orim12d 949 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐴 = 𝐵𝐵𝐴) → ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
2524con3d 149 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴)) → ¬ (𝐴 = 𝐵𝐵𝐴)))
26 simpl3 1231 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐶 ∈ ω)
27 simpl1 1227 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐴 ∈ ω)
28 nnmcl 7846 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐴 ∈ ω) → (𝐶 ·𝑜 𝐴) ∈ ω)
2926, 27, 28syl2anc 573 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐴) ∈ ω)
30 simpl2 1229 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → 𝐵 ∈ ω)
31 nnmcl 7846 . . . . . . . . 9 ((𝐶 ∈ ω ∧ 𝐵 ∈ ω) → (𝐶 ·𝑜 𝐵) ∈ ω)
3226, 30, 31syl2anc 573 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐶 ·𝑜 𝐵) ∈ ω)
33 nnord 7220 . . . . . . . . 9 ((𝐶 ·𝑜 𝐴) ∈ ω → Ord (𝐶 ·𝑜 𝐴))
34 nnord 7220 . . . . . . . . 9 ((𝐶 ·𝑜 𝐵) ∈ ω → Ord (𝐶 ·𝑜 𝐵))
35 ordtri2 5901 . . . . . . . . 9 ((Ord (𝐶 ·𝑜 𝐴) ∧ Ord (𝐶 ·𝑜 𝐵)) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ↔ ¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
3633, 34, 35syl2an 583 . . . . . . . 8 (((𝐶 ·𝑜 𝐴) ∈ ω ∧ (𝐶 ·𝑜 𝐵) ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ↔ ¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
3729, 32, 36syl2anc 573 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ↔ ¬ ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐵) ∈ (𝐶 ·𝑜 𝐴))))
38 nnord 7220 . . . . . . . . 9 (𝐴 ∈ ω → Ord 𝐴)
39 nnord 7220 . . . . . . . . 9 (𝐵 ∈ ω → Ord 𝐵)
40 ordtri2 5901 . . . . . . . . 9 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
4138, 39, 40syl2an 583 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
4227, 30, 41syl2anc 573 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
4325, 37, 423imtr4d 283 . . . . . 6 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴𝐵))
4443ex 397 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (∅ ∈ 𝐶 → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴𝐵)))
4544com23 86 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (∅ ∈ 𝐶𝐴𝐵)))
4619, 45mpdd 43 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → 𝐴𝐵))
4746, 19jcad 502 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) → (𝐴𝐵 ∧ ∅ ∈ 𝐶)))
485, 47impbid 202 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wne 2943  c0 4063  Ord word 5865  (class class class)co 6793  ωcom 7212   ·𝑜 comu 7711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-oadd 7717  df-omul 7718
This theorem is referenced by:  nnmword  7867  nnneo  7885  ltmpi  9928
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